standalone websearch extension (#23823)

## Summary

Add the extension-backed standalone `web.run` tool so Codex can call the
standalone search endpoint through the `codex-api` search client and
return its encrypted output to Responses.

- gate the new tool behind `standalone_web_search`
- install the extension in the app-server thread registry and hide
hosted `web_search` when standalone search is enabled for OpenAI
providers so the two paths stay mutually exclusive
- build search context from persisted history using a small tail
heuristic: previous user message, assistant text between the last two
user turns capped at about 1k tokens, and current user message

## Test Plan

- `cargo test -p codex-web-search-extension`
- `cargo test -p codex-api`
- `cargo test -p codex-core
hosted_tools_follow_provider_auth_model_and_config_gates`

codex-rs/codex-api/Cargo.toml

@@ -16,6 +16,7 @@ codex-utils-rustls-provider = { workspace = true }
 futures = { workspace = true }
 http = { workspace = true }
 reqwest = { workspace = true, features = ["json", "stream"] }
+schemars = { workspace = true }
 serde = { workspace = true, features = ["derive"] }
 serde_json = { workspace = true }
 thiserror = { workspace = true }